Extra dimensions and Lorentz invariance violation

TL;DR

This paper explores how interactions between photons and scalar fields from extra dimensions can modify photon dispersion relations, leading to energy-dependent velocities, and proposes using gamma-ray burst observations to constrain these effects.

Contribution

It introduces a model where photon interactions with internal space scalar fields cause Lorentz invariance violation and derives observational constraints from GRB time delays.

Findings

Photon dispersion relations are modified by scalar field interactions.

Photon group velocity depends on energy, causing time delays.

Experimental bounds from GRB observations constrain the model parameters.

Abstract

We consider effective model where photons interact with scalar field corresponding to conformal excitations of the internal space (geometrical moduli/gravexcitons). We demonstrate that this interaction results in a modified dispersion relation for photons, and consequently, the photon group velocity depends on the energy implying the propagation time delay effect. We suggest to use the experimental bounds of the time delay of gamma ray bursts (GRBs) photons propagation as an additional constrain for the gravexciton parameters.

Equations

$$S_{EM}=-\frac{1}{2}\underset{M}{\int }{d}^{D}x\sqrt{|g|}{F}_{MN}{F}^{MN},$$

$${\bar{S}}_{EM}=-\frac{1}{2}{\int }_{M_0}{d}^{D_0}x\sqrt{|{\tilde{g}}^{(0)}|}\left[\left(1-\mathcal{D}{\kappa }_0\psi \right)F_{\mu \nu }{F}^{\mu \nu }\right],$$

$${\partial }_{\nu }\left[\sqrt{-g}\left(1-\mathcal{D}{\kappa }_0\psi \right)F^{\mu \nu }\right]=0.$$

$$\nabla \cdot \mathbf{B}$$

$$\nabla \cdot \mathbf{E}$$

$$\nabla \times \mathbf{B}$$

$$\nabla \times \mathbf{E}$$

$$\mathbf{F}(\mathbf{k},\omega )$$

$$\mathbf{F}(\mathbf{x},\eta )$$

$$(1-n_{+}^2)E^{+}=0,(1-n_{(-)}^2)E^{-}=0,$$

$$n_{+}^2=\frac{k^2\left[1-\mathcal{D}{\kappa }_0\psi (1+q_z/k)\right]}{{\omega }^2\left[1-\mathcal{D}{\kappa }_0\psi (1+{\omega }_{\psi }/\omega )\right]}=n_{-}^2.$$

$$\frac{\partial k}{\partial \omega }\simeq \pm \left\{1+\frac{1}{2}\left[\frac{{\omega }_{\psi }^2-q_z^2}{4{\omega }^2}\right](\mathcal{D}{\kappa }_0\psi {)}^2\right\},$$

$$|\psi |\approx \frac{1}{\sqrt{\pi }\mathcal{D}}\sqrt{\left|\frac{\Delta t}{\Delta l}\right|}\frac{\omega }{m_{\psi }}{M}_{Pl},$$

$$\left|{\kappa }_0\psi \right|\approx \frac{10^{-13}\text{mbox}GeV}{m_{\psi }}.$$

$$\frac{d^2}{d{t}^2}\psi +(3H+\Gamma )\frac{d}{dt}\psi +m_{\psi }^2\psi =0,$$

$$\psi \approx CB(t)\cos (m_{\psi }t+\delta ).$$

$$|{\kappa }_0\psi |\sim 10^{-60}\frac{{\psi }_{in}}{M_{Pl}}{\left(\frac{M_{Pl}}{m_{\psi }}\right)}^{3/4}.$$

Full text

Lorentz invariance (LI) of physical laws is one of the corner stone of modern physics. There is a number of experiments confirming this symmetry at energies we can approach now. For example, on a classical level, the rotation invariance has been tested in Michelson-Morley experiments, and the boost invariance has been tested in Kennedy-Torhndike experiments reviews . Although, up to now, LI is well established experimentally, we cannot say surely that at higher energies it is still valid. Moreover, modern astrophysical and cosmological data (e.g. UHECR, dark matter, dark energy, etc) indicate for a possible LI violation (LV). To resolve these challenges, there are number of attempts to create new physical models, such as M/string theory, Kaluza-Klein models, brane-world models, etc. reviews .

In this paper we investigate LV test related to photon dispersion measure (PhDM). This test is based on the LV effect of a phenomenological energy-dependent speed of photon a98 ; ellis00 ; km01 ; sarkar ; myers ; piran03 ; jlm03 , for recent studies see Ref. MP06 and references therein.

The formalism that we use is based on the analogy with electromagnetic waves propagation in a magnetized medium, and extends previous works cfj90 ; jlm03 ; Tina1 . In our model, instead of propagation in a magnetized medium, the electromagnetic waves are propagating in vacuum filled with a scalar field $\psi$. LV occurs because of an interaction term $\mathrm{f}(\psi)F^{2}$ where $F$ is an amplitude of the electromagnetic field. Such an interaction might have different origins. In the string theory $\psi$ could be a dilaton field string ; damour-pol . The field $\psi$ could be associated with geometrical moduli. In brane-world models the similar term describes an interaction between the bulk dilaton and the Standard Model fields on the brane zhuk . In Ref. klp , such an interaction was obtained in $N=4$ super-gravity in four dimensions. In Kaluza-Klein models the term $\mathrm{f}(\psi)F^{2}$ has the pure geometrical origin, and it appears in the effective, dimensionally reduced, four dimensional action (see e.g. kubyshin ; GSZ ). In particular, in reduced Einstein-Yang-Mills theories, the function $f(\psi)$ coincides (up to a numerical prefactor) with the volume of the internal space. Phenomenological (exactly solvable) models with spherical symmetries were considered in Refs. melnikov . To be more specific, we consider the model which is based on the reduced Einstein-Yang-Mills theory GSZ , where the term $\propto\psi F^{2}$ describes the interaction between the conformal excitations of the internal space (gravexcitons) and photons. It is clear that the similar LV effect exists for all types of interactions of the form $\mathrm{f}(\psi)F^{2}$ mentioned above.

Obviously, the interaction term $\mathrm{f}(\psi)F^{2}$ modifies the Maxwell equations, and, consequently, results in a modified dispersion relation for photons. We show that this modification has rather specific form. For example, we demonstrate that refractive indices for the left and right circularly polarized waves coincide with each other. Thus, rotational invariance is preserved. However, the speed of the electromagnetic wave’s propagation in vacuum differs from the speed of light $c$. This difference implies the time delay effect which can be measured via high-energy GRB photons propagation over cosmological distances (see e.g. Ref. MP06 ). It is clear that gravexcitons should not overclose the Universe and should not result in variations of the fine structure constant. These demands lead to a certain constrains for gravexcitons (see Refs. GSZ ; iwara ). We use the time delay effect, caused by the interaction between photons and gravexcitons, to get additional bounds on the parameters of gravexcitons.

The starting point of our investigation is the Abelian part of D-dimensional action of the Einstein-Yang-Mills theory:

[TABLE]

where the D-dimensional metric, $g=g_{MN}(X)dX^{M}\otimes dX^{N}=g^{(0)}(x)_{\mu\nu}dx^{\mu}\otimes dx^{\nu}+a_{1}^{2}(x)g^{(1)}$, is defined on the product manifold $M=M_{0}\times M_{1}$. Here, $M_{0}$ is the $(D_{0}=d_{0}+1)$-dimensional external space. The $d_{1}$-dimensional internal space $M_{1}$ has a constant curvature with the scale factor $a_{1}(x)\equiv L_{Pl}\exp\beta^{1}(x)$. Dimensional reduction of the action (1) results in the following effective $D_{0}$-dimensional action GSZ

[TABLE]

which is written in the Einstein frame with the $D_{0}$-dimensional metric, $\tilde{g}^{(0)}_{\mu\nu}=(\exp d_{1}\bar{\beta}^{1})^{-2/(D_{0}-2)}g^{(0)}_{\mu\nu}$. Here, $\kappa_{0}\psi\equiv-\bar{\beta}^{1}\sqrt{(D_{0}-2)/d_{1}(D-2)}\ll 1$ and $\bar{\beta}^{1}\equiv\beta^{1}-\beta^{1}_{0}$ are small fluctuations of the internal space scale factor over the stable background $\beta^{1}_{0}$ ([math] subscript denotes the present day value). These internal space scale-factor small fluctuations/oscillations have the form of a scalar field (so called gravexciton GZ1 ) with a mass $m_{\psi}$ defined by the curvature of the effective potential (see for detail GZ1 ). Action (2) is defined under the approximation $\kappa_{0}\psi<1$ that obviously holds for the condition111In the brane-world model the prefactor $\kappa_{0}$ in the expression for $\kappa_{0}\psi$ is replaced by the parameter proportional to $M^{-1}_{EW}$ zhuk . Thus, the smallness condition holds for $\psi<M_{EW}$. $\psi<M_{Pl}$. $\kappa_{0}^{2}=8\pi/M^{2}_{Pl}$ is four dimensional gravitational constant, $M_{Pl}$ is the Plank mass, $\mathcal{D}=2\sqrt{d_{1}/[(D_{0}-1)(D-1)]}$ is a model dependent constant. The Lagrangian density for the scalar field $\psi$ reads: $\mathcal{L}_{\psi}=\sqrt{|\tilde{g}^{(0)}|}(-\tilde{g}^{\mu\nu}\psi_{,\mu}\psi_{,\nu}-m_{\psi}^{2}\psi\psi)/2$. For simplicity we assume that $\tilde{g}^{0}$ is the flat Friedman-Lemaitre-Robertson-Walker (FLRW) metric with the scale factor $a(t)$.

Let’s consider Eq. (2). It is worth of noting that the $D_{0}$-dimensional field strength tensor, $F_{\mu\nu}$, is gauge invariant.222Eq. (2) can be rewritten in the more familiar form $\bar{S}_{EM}=-(1/2)\int_{M_{0}}d^{D_{0}}x\sqrt{|\tilde{g}^{(0)}|}\bar{F}_{\mu\nu}\bar{F}^{\mu\nu}$ GSZ . The field strength tensor $\bar{F}_{\mu\nu}$ is not gauge invariant here. Secondly, action (2) is conformally invariant in the case when $D_{0}=4$. The transform to the Einstein frame does not break gauge invariance of the action (2), and the electromagnetic field is antisymmetric as usual, $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. Varying (2) with respect to the electromagnetic vector potential,

[TABLE]

The second term in the round brackets $\mathcal{D}\kappa_{0}\psi F^{\mu\nu}$ reflects the interaction between photons and the scalar field $\psi$, and as we show below, it is responsible for LV. In particular, coupling between photons and the scalar field $\psi$ makes the speed of photons different from the standard speed of light. Eq. (3) together with Bianchi identity (which is preserved in the considered model due to gauge-invariance of the tensor, $F_{\mu\nu}$ GSZ ) defines a complete set of the generalized Maxwell equations. As we noted, action (2) is conformally invariant in the $4D$ dimensional space-time. So, it is convenient to present the flat FLRW metric $\tilde{g}^{0}$ in the conformally flat form: $\tilde{g}^{0}_{\mu\nu}=a^{2}\eta_{\mu\nu}$, where $\eta_{\mu\nu}$ is the Minkowski metric.

Using the standard definition of the electromagnetic field tensor, $F_{\mu\nu}$, we obtain the complete set of the Maxwell equations in vacuum,

[TABLE]

where all operations are performed in the Minkowski space-time, $\eta$ denotes conformal time related to physical time $t$ as $dt=a(\eta)d\eta$, and an overdot represents a derivative with respect to conformal time $\eta$.

Eqs. (4) and (7) correspond to Bianchi identity, and since it is preserved, Eqs. (4) and (7) keep their usual forms. Eqs. (5) and (6) are modified due to interactions between photons and gravexcitons ($\propto\mathcal{\kappa}_{0}\psi$). These modifications have simple physical meaning: the interaction between photons and the scalar field $\psi$ acts as an effective electric charge $e_{eff}$. This effective charge is proportional to the scalar product of the $\psi$ field gradient and the ${\bf E}$ field, and it vanishes for an homogeneous $\psi$ field. The modification of Eq. (6) corresponds to an effective current ${\bf J}_{eff}$, which depends on both electric and magnetic fields. This effective current is determined by variations of the $\psi$ field over the time ($\dot{\psi}$) and space ($\nabla\psi$). For the case of a homogeneous $\psi$ field the effective current is still present and LV takes place. The modified Maxwell equations are conformally invariant. To account for the expansion of the Universe we rescale the field components as${\bf{B,E}}\rightarrow{\bf{B,E}}~{}a^{2}$ grasso .

To obtain a dispersion relation for photons, we use the Fourier transform between position and wavenumber spaces as,

[TABLE]

Here, ${\bf F}$ is a vector function describing either the electric or the magnetic field, $\omega$ is the angular frequency of the electro-magnetic wave measured today, and ${\bf k}$ is the wave-vector. We assume that the field $\psi$ is an oscillatory field with the frequency $\omega_{\psi}$ and the momentum $\bf q$, so $\psi({\bf x},\eta)=Ce^{i(\omega_{\psi}\eta-{\bf q}\cdot{\bf x})}\,,\quad C=\mbox{\rm const}\,$. Eq. (4) implies ${\bf B}\perp{\bf k}$. Without loosing of generality, and for simplicity of description we assume that the wave-vector ${\bf k}$ is oriented along the ${\bf z}$ axis. Using Eq. (7) we get ${\bf E}\perp{\bf B}$.

A linearly polarized wave can be expressed as a superposition of left (L, $-$) and right (R, $+$) circularly polarized (LCP and RCP) waves. Using the polarization basis of Sec. 1.1.3 of Ref. varshalovich89 , we derive $E^{\pm}=(E_{x}\pm iE_{y})/\sqrt{2}$. Rewriting Eqs. (4) - (7) in the components,333We have defined the system of 6 equations with respect to 6 components of the vectors ${\bf E}$ and ${\bf B}$. This system has non-trivial solutions only if its determinant is nonzero. From this condition we get the dispersion relation. The Faraday rotation effect is absent if the matrix has a diagonal form. for LCP and RCP waves we get,

[TABLE]

where $n_{+}$ and $n_{-}$ are refractive indices for RCP and LCP electromagnetic waves

[TABLE]

In the case when LI is preserved the electromagnetic waves propagating in vacuum have $n_{+}=n_{-}=n=k/\omega\equiv 1$. For the electromagnetic waves propagating in the magnetized plasma, $k/\omega\neq 1$, and the difference between the LCP and RCP refractive indices describes the Faraday rotation effect, $\alpha\propto\omega(n_{+}-n_{-})$ krall . In the considered model, since $n_{+}=n_{-}$ the rotation effect is absent, but the speed of electromagnetic waves propagation in vacuum differs from the speed of light $c$ (see also Ref. Cantcheff:2004dn for LV induced by electromagnetic field coupling to other generic field). This difference implies the propagation time delay effect, $\Delta t=\Delta l(1-\partial k/\partial\omega)$ ($\Delta l$ is a propagation distance), $\Delta t$ is the difference between the photon travel time and that for a ”photon” which travels at the speed of light $c$. Here, $t$ is physical synchronous time. This formula does not take into account the evolution of the Universe. However, it is easy to show that the effect of the Universe expansion is negligibly small.

Solving the dispersion relation as a square equation, we obtain

[TABLE]

where $\pm$ signs correspond to photons forward and backward directions respectively.

The modified inverse group velocity (11) shows that the LV effect can be measured if we know the gravexciton frequency $\omega_{\psi}$, $z$-component of the momentum $q_{z}$ and its amplitude $\psi$. For our estimates, we assume that $\psi$ is the oscillatory field, satisfying (in local Lorentz frame) the dispersion relation, $\omega_{\psi}^{2}=m_{\psi}^{2}+{\bf q}^{2}$, where $m_{\psi}$ is the mass of gravexcitons444To get physical values of the corresponding parameters we should rescale them by the scale factor $a$.. Unfortunately, we do not have any information concerning parameters of gravexcitons (some estimates can be found in GSZ ; iwara ). Thus, we intend to use possible LV effects (supposing it is caused by interaction between photons and gravexcitons) to set limits on gravexciton parameters. For example, we can easily get the following estimate for the upper limit of the amplitude of gravexciton oscillations:

[TABLE]

where for $\omega$ and $m_{\psi}$ we can use their physical values. In the case of GRB with $\omega\sim 10^{21}\div 10^{22}$Hz $\sim 10^{-4}\div 10^{-3}$GeV and $\Delta l\sim 3\div 5\times 10^{9}$y $\sim 10^{17}$sec the typical upper limit for the time delay is $\Delta t\sim 10^{-4}$sec MP06 . For these values the upper limit on gravexciton amplitude of oscillations is555We thank R. Lehnert to point that in addition of the time delay effect the Cherenkov effect could be used to constrain the electromagnetic field and $\psi$ field coupling strength cherenkov .

[TABLE]

This estimate shows that our approximation $\kappa_{0}\psi<1$ works for gravexciton masses $m_{\psi}>10^{-13}$GeV. Future measurements of the time-delay effect for GRBs at frequencies $\omega\sim 1-10$GeV would increase significantly the limit up to $m_{\psi}>10^{-9}$GeV. On the other hand, Cavendish-type experiments dvali ; cavendish ) exclude fifth force particles with masses $m_{\psi}\lesssim 1/(10^{-2}\mbox{cm})\sim 10^{-12}$GeV which is rather close to our lower bound for $\psi$ field masses. Respectively we slightly shift the considered mass lower limit to be $m_{\psi}\geq 10^{-12}\mbox{GeV}$. These masses considerably higher than the mass corresponding to the equality between the energy densities of the matter and radiation (matter/radiation equality), $m_{eq}\sim H_{eq}\sim 10^{-37}$GeV, where $H_{eq}$ is the Hubble ”constant” at matter/radiation equality. It means that such $\psi$-particles start to oscillate during the radiation dominated epoch (see appendix). Another bound on the $\psi$-particles masses comes from the condition of their stability. With respect to decay $\psi\to\gamma\gamma$ the life-time of $\psi$-particles is $\tau\sim(M_{Pl}/m_{\psi})^{3}t_{Pl}$ GSZ , and the stability conditions requires that the decay time should be greater than the age of the Universe. According this we consider light gravexcitons with masses $m_{\psi}\leq 10^{-21}M_{Pl}\sim 10^{-2}\mbox{GeV}\sim 20m_{e}$ (where $m_{e}$ is the electron mass).

As an additional restriction arises from the condition that such cosmological gravexcitons should not overclose the observable Universe. This reads $m_{\psi}\lesssim m_{eq}(M_{Pl}/\psi_{in})^{4}$ which implies the following restriction for the amplitude of the initial oscillations: $\psi_{in}\lesssim\left(m_{eq}/m_{\psi}\right)^{1/4}M_{PL}<<M_{Pl}$ iwara . Thus, for the range of masses $10^{-12}\mbox{GeV}\leq m_{\psi}\leq 10^{-2}\mbox{GeV}$, we obtain respectively $\psi_{in}\lesssim 10^{-6}M_{Pl}$ and $\psi_{in}\lesssim 10^{-9}M_{Pl}$. According to Eq. (A.3), we can also get the estimate for the amplitude of oscillations of the considered gravexciton at the present time. Together with the non-overcloseness condition, we obtain from this expression that $|\kappa_{0}\psi|\sim 10^{-43}$ for $m_{\psi}\sim 10^{-12}$GeV and $\psi_{in}\sim 10^{-6}M_{Pl}$ and $|\kappa_{0}\psi|\sim 10^{-53}$ for $m_{\psi}\sim 10^{-2}$GeV and $\psi_{in}\sim 10^{-9}M_{Pl}$. Obviously, it is much less than the upper limit (13). Note, as we mentioned above, gravexcitons with masses $m_{\psi}\gtrsim 10^{-2}$GeV can start to decay at the present epoch. However, taking into account the estimate $|\kappa_{0}\psi|\sim 10^{-53}$, we can easily get that their energy density $\rho_{\psi}\sim(|\kappa_{0}\psi|^{2}/8\pi)M_{Pl}^{2}m_{\psi}^{2}\sim 10^{-55}\mbox{g}/\mbox{cm}^{3}$ is much less than the present energy density of the radiation $\rho_{\gamma}\sim 10^{-34}\mbox{g}/\mbox{cm}^{3}$. Thus, $\rho_{\psi}$ contributes negligibly in $\rho_{\gamma}$. Otherwise, the gravexcitons with masses $m_{\psi}\gtrsim 10^{-2}$GeV should be observed at the present time, which, obviously, is not the case.

Additionally, it follows from Eq. (42) in Ref. GSZ that to avoid the problem of the fine structure constant variation, the amplitude of the initial oscillations should satisfy the condition: $\psi_{in}\lesssim 10^{-5}M_{Pl}$ which, obviously, completely agrees with our upper bound $\psi_{in}\lesssim 10^{-6}$GeV.

Summarizing we shown that LV effects can give additional restrictions on parameters of gravexcitons. First, we found that gravexcitons should not be lighter than $10^{-13}$GeV. It is very close to the limit following from the fifth-force experiment. Moreover, experiments for GRB at frequencies $\omega>1$GeV can result in significant shift of this lower limit making it much stronger than the fifth-force estimates. Together with the non-overcloseness condition, this estimate leads to the upper limit on the amplitude of the gravexciton initial oscillations. It should not exceed $\psi_{in}\lesssim 10^{-6}$GeV. Thus, the bound on the initial amplitude obtained from the fine structure constant variation is one magnitude weaker than our one even for the limiting case of the gravexciton masses. Increasing the mass of gravexcitons makes our limit stronger. Our estimates for the present day amplitude of the gravexciton oscillations, following from the obtained above limitations, show that we cannot use the LV effect for the direct detections of the gravexcitons. Nevertheless, the obtained bounds can be useful for astrophysical and cosmological applications. For example, let us suppose that gravexcitons with masses $m_{\psi}>10^{-2}$GeV are produced during late stages of the Universe expansion in some regions and GRB photons travel to us through these regions. Then, Eq. (A.3) is not valid for such gravexcitons having astrophysical origin and the only upper limit on the amplitude of their oscillations (in these regions) follows from Eq. (13). In the case of TeV masses we get $|\kappa_{0}\psi|\sim 10^{-16}$. If GRB photons have frequencies up to 1 TeV, $\omega\sim 1$TeV, then this estimate is increased by 6 orders of magnitude.

Acknowledgments

We thank G. Dvali, G. Gabadadze, A. Gruzinov, G. Melikidze, B. Ratra, and A. Starobinsky for stimulating discussions. T. K. and A. Zh. acknowledge hospitality of Abdus Salam International Center for Theoretical Physics (ICTP) where this work has been started. A.Zh. would like to thank the Theory Division of CERN for their kind hospitality during the final stage of this work. T.K. acknowledges partial support from INTAS 061000017-9258 and Georgian NSF ST06/4-096 grants.

.1 Appendix: Dynamics of Light Gravexcitons

In this appendix we briefly summarize the main properties of the light gravexcitons necessary for our investigations. The more detail description can be found in Refs. GSZ ; iwara .

The effective equation of motion for massive cosmological gravexciton666We have seen that the interaction between gravexcitons and ordinary matter (in our case it is 4D-photons) is suppressed by the Planck scale. Thus, gravexcitons are weakly interacting massive particles (WIMPs). is

[TABLE]

where $H\sim 1/t$ and $\Gamma\sim m_{\psi}^{3}/M_{Pl}^{2}$ are the Hubble parameter and decay rate ($\psi\to\gamma\gamma$) correspondingly. This equation shows that at times when the Hubble parameter is less than the gravexciton mass: $H\lesssim m_{\psi}$ the scalar field begins to oscillate (i.e. time $t_{in}\sim H^{-1}_{in}\sim 1/m_{\psi}$ roughly indicates the beginning of the oscillations):

[TABLE]

We consider cosmological gravexcitons with masses $10^{-12}\mbox{GeV}\leq m_{\psi}\leq 10^{-2}$GeV. The lower bound follows both from the fifth-force experiments and Eq. (13). The upper bound follows from the demand that the life-time of these particles (with respect to decay $\psi\to\gamma\gamma$) is larger than the age of the Universe: $\tau=1/\Gamma\sim\left(M_{Pl}/m_{\psi}\right)^{3}t_{Pl}\geq 10^{19}\mbox{sec}>t_{univ}\sim 4\times 10^{17}$ sec. Thus, we can neglect the decay processes for these gravexcitons. Additionally, it can be easily seen that these particles start to oscillate before $t_{eq}\sim H^{-1}_{eq}$ when the energy densities of the matter and radiation become equal to each other (matter/radiation equality). According to the present WMAP data for the $\Lambda$CDM model it holds $H_{eq}\equiv m_{eq}\sim 10^{-56}M_{Pl}\sim 10^{-28}\mbox{eV}$. Thus, considered particles have masses $m_{\psi}>>m_{eq}$ and start to oscillate during the radiation dominated stage. They will not overclose the observable Universe if the following condition is satisfied: $m_{\psi}\lesssim m_{eq}(M_{Pl}/\psi_{in})^{4}$, where $\psi_{in}$ is the amplitude of the initial oscillations at the moment $t_{in}$ (see Eq. (18) in Ref. iwara ).

Prefactors $C$ and $B(t)$ in Eq. (A.2) for considered light gravexcitons respectively read: $C\sim(\psi_{in}/M_{Pl})\left(M_{Pl}/m_{\psi}\right)^{3/4}$ and $B(t)\sim M_{Pl}\left(M_{Pl}t\right)^{-3s/2}$. Here, $s=1/2,2/3$ for oscillations during the radiation dominated and matter dominated stages, correspondingly. We are interested in the gravexciton oscillations at the present time $t=t_{univ}$. In this case $s=2/3$ and for $B(t_{univ})$ we obtain: $B(t_{univ})\sim t^{-1}_{univ}\approx 10^{-61}M_{Pl}$. Thus, the amplitude of the light gravexciton oscillations at the present time reads:

[TABLE]